What do X-ray powder diffraction patterns look like?

what information available in pattern? Powder patterns - what information available in pattern? 1. peak positions peak intensities - get crystal structure peak shape background structure Intensities give atom positions

Intensities X-rays scattered by electrons - electrons are in atoms Scattering power f of atom at  = 0° is no. electrons (atomic no.) x scattering power of one e—

Intensities Because of path length difference, scattering power decreases as increases Decrease is gradual since path length difference is small compared to  Atomic scattering factor f plotted as a function of (sin /. f values given in various tables, and as analytical functions

Intensities Now think of atoms in unit cells of a lattice Waves reflected (diffracted) from the same atoms at the same x, y, z positions in all unit cells will be in phase Thus, need to consider scattering in only one unit cell Out-of-phaseness depends on: a. relative positions of atoms xj, yj, zj b. diffraction angle  As before, the scattered waves out of phase But distances between the atoms much larger than between e— s

Intensities Wave from each atom has: amplitude - ƒ phase factor - exp (2πij) = exp (2πi(hxj + kyj + lzj) xj, yj, zj are positions h, k, l related to diffraction angle The scattering power for all atoms in unit cell obtained by adding up all scattered waves. This is the structure factor or structure amplitude

Intensities Since Fhkl is an amplitude Ihkl ~ |Fhkl|2 In general, F is imaginary ........ so F = A + iB and F*F = (A - iB) (A + iB)

Intensities Simple example calculation: Cu: Fm3m a = 3.614 Å Cu atoms in 4a - (000) (1/2 0 1/2) (1/2 1/2 0) (0 1/2 1/2) Equipoint 4a —> 4 atoms/cell —> 4 terms: Fhkl = ƒCu (e0 + eπi(h + l) + eπi(h + k) + eπi(k + l)) Since ei = cos  + i sin  (Euler's rule) Fhkl = 0 or 4ƒCu

Intensities Derivation of extinction rule for I centering: For every atom at (xj, yj, zj), there must be an atom at (xj + 1/2, yj + 1/2, zj + 1/2) Then Fhkl = ∑ ƒj {(exp (2πi(hxj + kyj + lzj) + exp (2πi(h(xj + 1/2) + k(yj + 1/2) + l(zj + 1/2)))} j = 1 N/2 Since eA + B = eA eB Fhkl = ∑ ƒj (exp (2πi(hxj + kyj + lzj)))(1 + eπi(h + k + l)) j = 1 N/2

Intensities Derivation of extinction rule for I centering Since eA + B = eA eB Fhkl = ∑ ƒj (exp (2πi(hxj + kyj + lzj)))(1 + eπi(h + k + l)) j = 1 N/2 Every term in sum contains 1 + eπi(h + k + l) = 1 + cos (π(h + k + l)) + i sin (π(h + k + l)) —> 0 when h + k + l = odd number Extinction rule for I centering: (hkl), h + k + l = 2n

Intensities A slightly more complex structure: HoZn2 is I 2/m 2/m 2/a, with a = 4.456 ± 1, b = 7.039 ± 3, c = 7.641 ± 5Å Ho in 4e (0,1/4,z) (0,3/4,z) + I , z = 0.5281 ± 4 Zn in 8h (0,y,z) (0,y,z) (0,1/2 + y,z) (0,1/2-y,z) + I, y = 0.0410 ± 9, z = 0.1663 ± 8 F(hkl) = fHo (exp (2πi(k/4 + l(0.5281))) + exp (2πi(3/4k + l(0.4719))) + exp (2πi(1/2h + 3/4k + l(0.0281))) + 1 more term) + fZn (exp (2πi(k(0.0410) + l(0.1663))) + 7 more terms)

Intensities Now Ihkl = scale factor • p • LP • A • |Fhkl|2 • e–2M p = multiplicity accounts for differing probabilities that symmetry equivalent planes (hkl) will reflect Example - cubic {100} = (100), (010), (001), (100), (010), (001) p = 6 {110} = (110), (101), (011),, (110), (101), (011), (110), (101), (011), (110), (101), (011) p = 12

Intensities Now Ihkl = scale factor • p • LP • A • |Fhkl|2 • e–2M LP = Lorentz-polarization factor P accounts for polarization state of incident beam (in most powder x-ray diffractometers, unpolarized) Lorentz factor corrects for geometrical broadening of reflections as 2 increased

Intensities Now Ihkl = scale factor • p • LP • A • |Fhkl|2 • e–2M Lorentz factor corrects for geometrical broadening of reflections as 2 increased Which reflection is more intense?

Intensities Now Ihkl = scale factor • p • LP • A • |Fhkl|2 • e–2M LP = Lorentz-polarization factor LP = (1 + cos2 2q)/sin2 q cos q

Intensities Now Ihkl = scale factor • p • LP • A • |Fhkl|2 • e–2M A = absorption factor In standard X-ray powder diffractometers, when specimen is dense and thick, A is considered constant over all 2

Intensities Now Ihkl = scale factor • p • LP • A • |Fhkl|2 • e–2M(T) e–2M(T) = temperature factor (also called Debye-Waller factor) 2M(T) = 16π2 m(T)2 (sin q)2/l2 m2 = mean square amplitude of thermal vibration of atoms direction normal to planes (hkl) I(high T) e–2M(high T) I(low T) e–2M(low T) = 1 e2M(high T) - 2M(low T)

Intensities —> crystal structure So, OK, how do we do it? Outline of procedure: Measure reflection positions in x-ray diffraction pattern - index, get unit cell type and size, possible space groups Measure density, if possible, to get number formula units/unit cell (N) density = N x formula wt/cell volume x Avogadro's no. Measure reflection intensities, get F-values, calculate electron density distribution from

Intensities —> crystal structure Electron density distribution tells where the atoms are anthracene (XYZ) is plotted and contoured to show regions of high electron density

Intensities —> crystal structure But WAIT!!! Ihkl = K |Fhkl|2 = K Fhkl* x Fhkl = K (Ahkl - iBhkl) (Ahkl + iBhkl) = K (Ahkl2 + Bhkl2) √Ihkl/K = √(Ahkl2 + Bhkl2) So, can't use Ihkls directly to calculate Fhkls and (XYZ)!! Many techniques for using Ihkls to determine atom positions have been developed, most of which, at some stage, involve formulating a model for the crystal structure, and then adjusting it to fit the intensity data

what information available in pattern? Powder patterns - what information available in pattern? 1. peak positions peak intensities peak shape - peak broadening background structure Two effects

Peak broadening Here, have set of planes reflecting in-phase at  (Bragg condition) X-rays at non- angles won't reflect; every reflected ray has a mate deeper in the crystal which is 180° out of phase with it - reflection is narrow Small crystallite size broadens reflections - becomes significant below 1 micron

Peak broadening Microstrain & chemical inhomogeneity distort the structure so that interplanar distances not constant - vary a little from an average value - broadens peaks

Peak broadening If broadening due to small crystallite size only, simple technique to determine that size, L, from the breadth, B Scherrer eqn.: Bsize = (180/π) (K/ L cos ) (K - 0.9, usually) Btot2 = Binstr2 + Bsize2 Must subtract broadening due to instrument. Measure peak width from suitable standard (ex: LaB6 - NIST SRM 660)

Peak broadening If broadening due to small crystallite size only, simple techique to determine that size, L, from the breadth, B Scherrer eqn.: Bsize = (180/π) (K/ L cos ) (K - 0.9, usually) Btot2 = Binstr2 + Bsize2 104Å Bsize = (180/π) (1.54/ 104 cos 45°) = 0.0125° 2 103Å Bsize = 0.125° 2 102Å Bsize = 1.25° 2 10Å Bsize = 12.5° 2

Peak broadening If broadening due to small crystallite size & microstrain, simple technique to determine size, L, & strain, <>, from the breadth, B Williamson-Hall method: strain broadening - Bstrain = <>(4 tan ) size broadening - Bsize = (K/ L cos ) (Bobs − Binst) = Bsize + Bstrain (Bobs − Binst) = (K / L cos ) + 4 <ε>(tan θ) (Bobs − Binst) cos  = (K / L) + 4 <ε>(sin θ)

Peak broadening If broadening due to small crystallite size & microstrain, simple technique to determine size, L, & strain, <>, from the breadth, B Williamson-Hall method: (Bobs − Binst) cos  = (K / L) + 4 <ε>(sin θ)

Peak broadening If broadening due to small crystallite size & microstrain, can use other more complex methods for analysis: Warren-Averbach double Voigt x-ray tracing

what information available in pattern? Powder patterns - what information available in pattern? 1. peak positions peak intensities - get crystal structure peak shape background structure Obvious structural information in back-ground - due to non-crystalline character of material (disorder)